To evaluate $P(c)$ for $P(x) = 2x^7 + x^2 - 4$ and $c = -1$ using synthetic division and the Remainder Theorem, we proceed with the following steps:

Step 1: Write down the coefficients of $P(x)$

The polynomial is:

$P(x) = 2x^7 + x^2 - 4$

Notice that there are missing terms in the polynomial (e.g., no $x^6, x^5, x^4, x^3, x^1$ terms). We need to account for these missing terms by adding zero coefficients:

$P(x) = 2x^7 + 0x^6 + 0x^5 + 0x^4 + x^2 + 0x^1 - 4$

Thus, the coefficients are:

$2, 0, 0, 0, 1, 0, 0, -4$

Step 2: Set up the synthetic division

We will divide $P(x)$ by $x + 1$, which corresponds to $c = -1$. For synthetic division, we use $c = -1$ and the coefficients:

$\text{Coefficients: } 2, 0, 0, 0, 1, 0, 0, -4$

Step 3: Perform the synthetic division

Let's perform the synthetic division with $c = -1$:

-1 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & -4 \\ & & -2 & 2 & -2 & 2 & -3 & 3 & -3 \\ \hline & 2 & -2 & 2 & -2 & 3 & -3 & 3 & -7 \\ \end{array}$$ ### Step 4: Interpretation of the result The final number in the bottom row is the remainder when dividing by $$x + 1$$. This remainder is the value of $$P(-1)$$. $$P(-1) = -7$$ ### Step 5: Conclusion The value of $$P(-1)$$ is $$-7$$, which is the remainder from the synthetic division. Therefore: $$P(-1) = -7$$ --- Would you like a detailed breakdown of any specific part of the synthetic division process? Here are some related questions for further exploration: